Timeline for In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?
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| Nov 23, 2020 at 15:43 | vote | accept | Arun Debray | ||
| Nov 23, 2020 at 6:09 | comment | added | Urs Schreiber | The character map and the AHSS are tools to break down any generalized cohomology into ordinary cohomology with a sequence of conditions and identifications imposed. It is in this way that, conversely, a web of flux quantization conditions on superficially ordinary cohomology classes may be unified and thus explained by charge quantization in a single but generalized cohomology theory which enforces them all. The underlying ordinary cohomology classes carved out thereby connect the generalized cohomology in particular to de Rham classes, but are not an end in themselves in the present context. | |
| Nov 23, 2020 at 6:07 | comment | added | David Roberts♦ | tl;dr "Hypothesis H sees the Hořava-Witten Green-Schwarz mechanism in the presence of M5-branes ... [which does not follow] from flux quantization in just ordinary cohomology (nor in K-theory, for that matter)." is a perfectly good statement, and is backed up by the rest of what you wrote. Just so that time-poor people without a focus on this can see an answer immediately. | |
| Nov 23, 2020 at 6:02 | comment | added | Urs Schreiber | If C-field flux were quantized in ordinary cohomology, there'd be no need for DFM to discuss "models" of it, namely non-ordinary cohomology theories whose structure enforces the peculiar nature of C-field flux. From plain ordinary cohomology in 11d evidently K-theory does not follow, or else the latter were pointless. Ordinary cohomology quantizes non-interacting abelian gauge flux and nothing else, that does't make an M-theory. It's extra conditions on top of superficially ordinary cohomology classes which are the content of DMW's old argument, as explained in the reply's first section. | |
| Nov 23, 2020 at 6:01 | comment | added | Urs Schreiber | The very first paragraphs of answer highlight that subquotients of ordinary cohomology groups are not ordinary cohomology anymore -- or else you'd conclude from the AHSS that everything is ordinary cohomology. There is a single flux quantization condition enforced by ordinary integral cohomology, namely integrality of the charge, and evidently that's not the subtle web of conditions in question here. | |
| Nov 22, 2020 at 20:57 | comment | added | David Roberts♦ | This has buried the lede somewhat. "Certainly none of these effects [in the previous paragraph] follows from flux quantization in just ordinary cohomology (nor in K-theory, for that matter)." seems to be the answer to the actual question. | |
| Nov 22, 2020 at 17:17 | history | edited | Urs Schreiber | CC BY-SA 4.0 |
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| Nov 22, 2020 at 16:26 | history | edited | Urs Schreiber | CC BY-SA 4.0 |
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| Nov 22, 2020 at 16:04 | history | edited | Urs Schreiber | CC BY-SA 4.0 |
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| Nov 22, 2020 at 15:31 | history | edited | Urs Schreiber | CC BY-SA 4.0 |
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| Nov 22, 2020 at 15:15 | history | answered | Urs Schreiber | CC BY-SA 4.0 |